1. On a connectedness theorem for a birational transformation at a simple point. [Mur58b]

The famous connectedness theorem of Oscar Zariski (1899–1986) states that the total transform of a normal point by a birational morphism is connected and in [Mur58b] it is shown that for a birational morphism the total transform of a simple point is rationally connected (i.e., any two points can be joined by a chain of rational curves). This result is on the one hand more precise than Zariski’s, but on the other hand more restricted because it applies to a simple point instead of to a normal point. Murre’s result “explains”, for instance, the well-known fact that a rational map of a variety to an abelian variety is regular at simple points. Grothendieck mentioned this fact in [13, Bourbaki Sem. 190, page 26] and asked Jacob if he could extend his result to mixed characteristic which he could not at that time. However, Wei-Liang Chow (1911–1995) in [7] answered Grothendieck’s question by proving the theorem in full generality.

In the subsequent paper [Mur58a], using his connectedness theorem, Jacob extended the notion of intersection multiplicity in the case of proper intersections, but also for maximal connected unions of intersections, solving a question posed by A. Weil in [30, p. 249].

2. On contravariant functors from the category of preschemes over a field into the category of abelian groups (with an application to the Picard functor).[Mur64]

In [13, Sem.Bourbaki 232] Grothendieck constructed the Picard scheme for a scheme projective over a base scheme. He first defines the Picard functor and then goes on to shows its representability. His method for the construction of the Picard scheme is close to the method used originally in Matsusaka’s construction of the Picard variety. However, Grothendieck replaced Chow varieties by Hilbert schemes. This uses in an essential way the projectivity over the base and so it leaves open the case of a scheme proper over a field. In the present article this gap has been filled, by constructing the Picard scheme for a scheme proper over a field. In rough outline the proof proceeds as follows. First, using all the fundamental tools created by Grothendieck in his previous Bourbaki talks (e.g. his theory of pro-representability, his comparison theorem and his famous existence theorem), necessary and sufficient conditions are established for the representability of a contravariant functor from the category of preschemes over a field to the category of abelian groups. Next it is shown that these necessary and sufficient conditions are satisfied by the Picard functor for a scheme proper over a field.

Further developments. In the fundamental paper [1] Michael Artin created the theory of algebraic spaces which “settled” in a certain sense the problem of representability of functors over an arbitrary base. In general one can not expect representability by a scheme and Artin had the idea to use instead algebraic spaces. Under certain natural conditions Artin could show that such a functor (over a fixed base) is indeed representable by an algebraic space. Since, as also shown by Artin, over a field algebraic spaces endowed with a group structure are in fact schemes, this gives an improvement of [Mur64].

3. The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme. [GM71]

Mumford in [22] studied the fundamental group of a normal point on a complex algebraic surface by considering a small tubular neighborhood of a suitable divisorial resolution of the singularity. As has been pointed out by Grothendieck in [14, VIII section 3] it is of interest to study this also in the abstract setting and this is the purpose of the present monograph, written jointly with Grothendieck. However, the final result is less nice and precise than Mumford’s result. In this abstract algebraic case one has to work within formal neighborhoods of the exceptional divisor, and with the algebraic fundamental group. Moreover, one has to restrict to tame coverings. Hence there is not a characterization of simple points in terms of this fundamental group as in the complex situation as Mumford showed. See also the notes [Mur67] of lectures given at the Tata institute.

4. Algebraic equivalence modulo rational equivalence on a cubic threefold. [Mur72]

One of the most striking results in 1971 in the field of algebraic geometry was the proof by Clemens and Griffiths of the non-rationality of the non-singular cubic threefold [8], solving a famous problem of classical Italian geometry. The crucial tool in their proof was the intermediate jacobian associated to any odd-dimensional cohomology group of a nonsingular projective variety. In general this is an analytic torus constructed via Hodge theory. In the present case of the cubic threefold one associates this torus to the middle cohomology and here this is in fact an abelian variety. It has a geometric flavor since it is determined by the family of lines on the threefold. In the non-rationality proof these algebraic cycles of codimension two play a crucial role. Since, as discovered earlier by Mumford and Griffiths, divisors and algebraic cycles of codimension \(\ge 2\) behave fundamentally different with respect to various equivalence relations, in this setting a natural question arose: what is the nature of the Abel–Jacobi equivalence relation?¹ In the present article that question was answered: for the cubic threefold the group of one dimensional cycles which are algebraically equivalent to zero modulo rational equivalence maps injectively to the intermediate jacobian. The method of proof is entirely algebraic and it works over every algebraically closed field of characteristic different from \(2\). In this algebraic setting the intermediate jacobian has to be replaced by a so-called Prym variety which itself is a principally polarized abelian variety (see also [Mur73] which is discussed in 5 below). To give an idea how the Prym variety enters the proof, first fix a (general) line \(L\) on the cubic \(X\) and a plane \(N\) not meeting \(L\) in the ambient 4-dimensional projective space. By taking the span \(M_p=\langle p,L\rangle \) of the points \(p\in N\) with \(L\), the intersections \(C_p=M_p\cap X\) form a family of conics (on \(X\)) and hence a conic bundle over \(N\). Over some curve \(C\) in \(N\) these conics degenerate into two lines (both meeting \(L\)). On the other hand we consider the curve \(C’ \) of the set of all the lines meeting the fixed line \(L\). This defines an involution of \(C’\) over \(C\) by interchanging these two lines and hence an involution of the level of the jacobian. This is the Prym involution and its ”kernel”, the corresponding Prym variety, takes the place of the intermediate jacobian of \(X\).

5. Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford. [Mur73]

Mumford has made a precise systematic study of Prym varieties in his paper [23]. Since the intermediate jacobian of a cubic threefold has an interpretation as a Prym variety (see the previous paper 4 and the references given there), one could in this way get a proof (in characteristic zero) of the non-rationality of the cubic threefold. Using Mumford’s results and methods from [Mur72], and working with étale cohomology instead of singular cohomology the transcendental methods could be replaced by algebraic ones. Hence the non-rationality of the non-singular cubic threefold holds over every algebraically closed field of different from \(2\).

Further developments: In 1977 there appeared a truly beautiful paper [5] by Arnaud Beauville (born 1947), were he made a profound study of quadric bundles and their corresponding Prym varieties and which implied the results of [Mur73] as a special case.

6. The Hodge conjecture for fourfolds admitting a covering by rational curves. [CM78c]

In this article, written jointly with A. Conte, the Hodge \((2,2)\) conjecture for a nonsingular complex projective fourfold is shown in case there exists a covering family of rational curves. In particular this applies to nonsingular cubic quartic and quintic fourfolds and also for quadric bundles of dimension \(4\). More examples were given in [CM80]

7. On the Chow groups of certain types of Fano threefolds. [BM79]

In this joint article with S. Bloch the Chow groups of codimension two cycles on three types of Fano threefolds \(X\) are investigated, namely quartic threefolds in \(\mathbb P ^4\), intersections of a cubic and a quadric in \(\mathbb P ^5\), and the intersections of three quadrics in \(\mathbb P^6 \); more precisely, we study their subgroups of rational equivalence classes of cycles algebraically equivalent to zero. These groups are shown to be weakly representable in the sense that they can be dominated by the jacobian of a curve via a correspondence between the curve and the variety in question² . In fact, they are parametrized by the points of a generalized Prym variety, i.e., a principally polarized abelian variety associated to a pair consisting of a jacobian variety of a curve and an endomorphism satisfying a quadratic equation.³ This result is greatly inspired by the beautiful paper [28] where such varieties (over \(\mathbb C\)) are considered via intermediate jacobians. Another important ingredient comes from Bloch’s article [6], where it is shown that there exists a map between the torsion groups of the codimension two cycles to the third étale cohomology group. In the present situation these morphisms are isomorphisms. Using this, certain questions on the Chow groups can be reduced to known results in étale cohomology.

8. Applications of algebraic \(K\)-theory to the theory of algebraic cycles. [Mur85b]

In this article it is shown that for codimension two cycles algebraically equivalent to zero there exists a universal regular map to an abelian variety (such a map is called regular if for every algebraic family \( T\) of such cycles the composite of the family and the map is itself a rational map - in the sense of algebraic geometry - from \( T\) to the abelian variety). Over the complex numbers this abelian variety is the image by the Abel-Jacobi map of the algebraic cycles of this kind in the intemediate jacobian.

The theorem answers a question of Mumford for such cycles [16, p. 143]. The proof uses results of Hiroshi Saito, Bloch and Bloch-Ogus in combination with the theorem of Merkurjev–Suslin from algebraic K-theory. In 2018 Bruno Kahn found a mistake in the proof of one the lemmas and fixed it in [18].

9. Abel–Jacobi equivalence versus incidence equivalence for algebraic cycles of codimennsion two. [Mur85a]

As discussed in 4, cycles of codimension \(j\) algebraically equivalent to zero are said to be Abel–Jacobi equivalent to zero if they are in the kernel of the Abel-Jacobi map to the \(j\)-th intermediate Jacobian. On the other hand there is also the subgroup of the cycles incident equivalent to zero, a concept introduced by P. Griffiths in [12]. The importance of this concept has also been stressed by Grothendieck in [15]. Griffiths proved that incidence equivalence is coarser than Abel–Jacobi equivalence and raised the question whether the two differ at most by an isogeny. For the case of codimension two cycles this question of Griffiths is answered in the positive as shown in this article. The main tool of the proof is Hodge theory and so the result is only valid over \(\mathbb C\).

10. On the motive of an algebraic surface. [Mur90]

One of Grothendieck’s reasons to create a theory of motives was to understand better the similarity of the cohomology groups of an algebraic variety in the different settings (classical étale, crystalline). He focussed on homological motives, i.e., motives built from algebraic cycles modulo homological equivalence (or rather, numerical equivalence, but this depends on a conjecture). His construction works for every adequate equivalence relation. Jacob in this article, inspired by works of Yuri Manin and Christophe Soulé, and by discussions with Spencer Bloch, turned to Chow motives, i.e., motives built using rational equivalence instead of homological equivalence. As he states in the introduction of loc. cit.: ”It is my conviction that Chow motives are more suited for the study of the Chow groups”. In Grothendieck’s theory one uses the cohomological Künneth components \( p^j (X) \) of the diagonal of the variety \(X\), but for Chow motives one replaces these with the finer Chow–Künneth components \( \text {ch}^j (X) \) belonging to the Chow group of \(X\times X\) and whose classes give the usual Künneth components. These components act as projectors. For every nonsingular projective variety \(X\) of dimension \(d\) (defined over an arbitrary field) one has the two trivial projectors \( \text {ch}^0(X) = e \times X \) and dually \( \text {ch}^{2d}(X) = X \times e \), where \(e\) is a (rational) point on \(X\). In the present paper for every such \(X\) the next Chow–Künneth projector \(\text {ch}^1(X) \) and its dual \( \text {ch}^{2d-1}(X) \) are constructed. The construction of \( \text {ch}^1 \) is based upon the work of A. Weil and S. Lang on the Picard and Albanese variety. Using a linear section of the variety one constructs first a morphism from the Picard variety to the Albanese variety which turns out to be an isogeny. An inverse of this isogeny gives a morphism from the Albanese to the Picard variety and therefore by the Weil–Lang theory one obtains a divisor class in the product \( X \times X \). This is a refined algebraic version of the \( \Lambda _{d-1}(X) \)-class from the Lefschetz decomposition valid for cohomology. These ingredients allow to contruct the Chow-Künneth projector \( \text {ch}^ 1 (X) \), and its transpose yields \( \text {ch}^{2d-1} (X) \). The corresponding Chow motives are the motivic versions of the Picard and Albanese variety (similarly, in the case of a curve \(C\) the Chow motive \( \text {ch}^1( C) \) is the motivic version of the jacobian variety of the curve \(C\)).

In the case of a surface this yields a complete decomposition of the diagonal since for the remaining \( \text {ch}^2(S) \) one can take the difference of the diagonal with the previous projectors. The surface \(S\) is now as Chow motive completely decomposed into five Chow motives \( \text {ch}^j(S) \) each of which not only gives back the \(j\)-th cohomology but on the Chow level this gives a filtration, the so-called Bloch–Beilinson filtration.⁴

In his beautiful survey paper in the 1991 Seattle conference Tony Scholl [27] has given a more modern description (and some slight improvement) of this paper. See also paper 13.

11. Motivic decomposition of abelian schemes and Fourier transform. [DM91]

Already in 1974 Shermenev – a student of Manin - has shown, by intricate arguments, that the Chow motive \(\text {ch} (A)\) of an abelian variety \(A\) over an algebraically closed field can be completely decomposed. In this paper, joint with C. Deninger, the existence of a canonical motivic decomposition is proven, not only for an abelian variety over a field but also for an abelian scheme \(A\) over a base which is itself over a field. For this purpose the theory of Chow motives is extended to a theory of relative Chow moties over a given base. A. Beauville has used the work of S. Mukai on the Fourier transform to study the decomposition of the Chow groups of an abelian variety under the endomorphism of multiplication by \(n\). Similar methods can be used on abelian schemes. Next, the article [20] by D. Lieberman⁵ then yields the required decomposition of the diagonal.

Further developments. K. Künnemann in [19] has extended these methods in a nice way and obtained a complete Lefschetz theory for Chow motives of an abelian scheme.

12. On a conjectural filtration on the Chow groups of an algebraic variety. Parts I,II. [Mur93a, Mur93b]

One of the famous standard conjectures is that the Künneth decomposition of the diagonal of a nonsingular projective algebraic variety \(X\) can be given by algebraic cycles on the product \(X \times X\). Based on some examples Jacob suggested an even stronger conjecture namely that those required cycle classes can in fact be lifted to rational equivalence and then give an orthogonal decomposition of the diagonal in the Chow ring, i.e., that there is not merely a Künneth decomposition but a Chow–Künneth decomposition which gives a filtration on the Chow groups (see the paper 10 above).

In [Mur93a] a number of conjectures were announced describing how this filtration should be determined by those Chow motives. In [Mur93b] part of these conjectures were verified in a (rather limited) number of cases. In particular for surfaces these are true (see the paper 10); moreover they are also true for threefolds of the type \(S \times C\) where \(S\) is a surface and \(C\) is a curve. For abelian surfaces part of the conjectures is true and the remaining part is equivalent to a conjecture of A. Beauville.

To this day the evidence for the truth of the conjectures is still meagre (the cornerstone in this story remains of course the famous Standard Conjecture itself about the algebraicity of the Künneth components of the diagonal themselves - the so called standard conjecture CK - for which unfortunately no progress has been obtained for a long time).

In his beautiful lecture in the Seattle Conference on Motives (1991) Uwe Jannsen [17] discussed these conjectures and he showed that they are equivalent with the so-called Bloch–Beilinson conjectures (in one of their forms) and that the corresponding filtrations (if they exist) also coincide.

13. On the transcendental part of the motive of a surface. [KMP07]

This is a joint paper with Bruno Kahn and Claudio Pedrini. A warning is in order: in this paper Voevodsky’s⁶ convention is used which is homological rather than cohomological and uses a covariant version of Chow motives which is the dual of Grothendieck’s contravariant (cohomological) version. In all other papers of Jacob he adhered to Grothendieck’s convention.

Recall that in the paper 10 a complete Chow-Künneth decomposition of the Chow motive of a surface \(S\) was given. Here the mysterious middle motive \( \text {ch}_2 (S) \) is further decomposed into an obvious algebraic part (carrying the algebraic cohomology) and a transcendental part \(t(S)\). The algebraic part is a sum of twisted Tate motives. However, the transcendental part is the truly mysterious part carrying the transcendental cohomology as well as the Albanese kernel. After the introduction there are three sections, each written by a different author. In the first part (written by Jacob) the validity of his conjectures is investigated for the motive of the product of two surfaces. In the second part (written by C. Pedrini) the birational motive of a surface is treated as well as the relation between Bloch’s conjecture and the so-called finite dimensionality conjecture by Kimura–O’Sullivan. Finally, in the last part (written by B. Kahn) again birational motives of a surface are studied and also the step is made from passing from pure motives to the triangulated category of Voevodsky.

References

Publications of J.P. Murre